Analytic Insights into an Adapted Algorithm for the Score-Based Secretary Problem (Journal of Mathematical Study, 2024)
In this paper, we study some basic analytic properties of a sequence of functions $\{S^{μ,σ}_n\}$ that is directly derived in an adaptive algorithm originating from the classical score-based secretary problem. More specifically, we show that: 1. the uniquenessof maximum points of the function sequence $\{S^{μ,σ}_n\};$ 2. the maximum point sequenceof $\{S^{μ,σ}_n\}$ monotone increases to infinity as $n$ tends to infinity. All of the proofs areelementary but nontrivial.